This short monograph by the late Soviet scientist Maslov is a book on elementary logic but it is not, by any means, an elementary book. Untimely death prevented the author from completing the work; this is apparent at many points in the text.

The book is arranged in three parts of the same length. Part 1 is entitled Mathematics of Calculi. Elementary logic is mainly devoted to the analysis of deduction using recursive function theory. It is well known that recursion can be attacked from various points of view. This is the essence of Church’s Thesis and the variety of perspectives ultimately simplifies the process. The proofs of some theorems may be difficult or easy depending upon the approach. But one must first justify Church’s Thesis, and this is a very tedious task.

Maslov uses only Post canonical systems, which are not in current use among Western scientists (although Post systems are similar to general grammars). Enumerable sets, normal forms for the derivations, dependence on the alphabet, universal systems, and algorithmic undecidability are outlined in 30 pages. The proofs of the involved theorems are far from complete; certainly the beginner will lose the way. (The exercises, often very difficult and without explanations, will be of little value.) But the experienced reader will appreciate the virtuoso performance of the author.

The last paragraph in Part 1 is devoted to the introduction of probability, through which one may investigate Post systems as Markov chains.

Part 2 is called Horizontal Modeling. In this part, under the name Economic Systems, and again from the viewpoint of Post systems, the author examines some of the problems for which Western experts use Petri nets. First, the author explains, through case studies, how to use inference rules to formalize the concept of an economic system with control centers, resources, and exchanges of resources between centers. The reader familiar with Petri nets will identify control centers with places, resources with markings, and rules of exchange with firing rules. Second, he deals with a variant, in terms of Post systems, of the construction of the coverability graph. (Caution: Maslov says “decidability” instead of “coverability”; this may be the cause of serious mistakes.) Finally, he investigates the ability of probabilistic Post systems to formalize evolutionary processes. Although formal grammars have been used in biology, Maslov instead follows the well-known analysis (by G. Hardy) of the stability of genotypes frequencies.

Part 3 is called Vertical Modeling. Here the author discusses mechanical theorem proving, undoubtedly the chapter the reader has been waiting for from the beginning. By a fine analysis of Gentzen’s logical calculus and of the analytical calculus of indefinite integrals, the author shows why the key is the elimination of the cut rules, that is, of rules such that for a fixed conclusion there can be an infinite set of corresponding lists of premises. Methods for such elimination are too briefly outlined. In particular the reader will be rather disappointed by the treatment of the inverse method, with which Maslov has set up his scientific reputation. (However, in the foreword, the reader is told that the author was not pleased with the present state of the text and had planned on rewriting it.)

The book ends with a fascinating analysis of nondeterminism, which links guessing with the opposing activities of the two cerebral hemispheres.

This is a challenging book; it does not observe the standards of mathematical writing. Proofs are only outlined, and philosophical reflections often replace them. So the book must not be recommended to the beginner, but certainly the experienced reader will enjoy Maslov’s personal description of the logic scenery.

References are not in the alphabetical order but follow the order of the first mention in the text. The absence of an index is sometimes troublesome: on p. 86 one encounters “the function I_{(A,Q) (p)” with no other explanation. The reader must remember that such notation was initially used on p. 58.}